Efficient Domination and Polarity Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) Universität Rostock Fakultät für Informatik und Elektrotechnik vorgelegt von Ragnar Christopher Nevries Gutachter: Prof. Dr. Andreas Brandstädt, Institut für Informatik, Universität Rostock Prof. Dr. Peter Widmayer, Institut für Theoretische Informatik, ETH Zürich Prof. Dr. Jing Huang, Mathematics and Statistics, University of Victoria eingereicht am 5. Februar 2014 verteidigt am 25. Juli 2014 Abstract This thesis considers Efficient Domination, Efficient Edge Domination, Polar- ity, and Monopolarity, graph problems that ask for a vertex or edge subset that is a packing and a covering at the same time. Efficient Domination seeks for an independent vertex subset D such that all other vertices have exactly one neighbor in D. Here, packing means that the vertices of D must not be too close to each other and, in contrast, covering requires that they have to be near to the other vertices. Efficient Edge Domination is the edge version of Efficient Domination. Polarity asks for a vertex subset that induces a complete multipartite graph—the packing aspect—and that contains a vertex of every induced P3—the covering aspect. Monopolarity is the special case of Polarity where the complete multipartite graph has to be edgeless. Since all these problems are NP-complete in general, for each problem a lot of effort has been put into separating the graph classes on which the problem remains NP-complete from those that admit an efficient algorithm. This thesis pursues both directions. On the one hand, we introduce a framework for our NP-completeness proofs and use it to sharpen known results for all mentioned problems. On the other hand, we reveal new tractable cases: Efficient Domination As we figure out that the problem is NP-complete on F -free graphs whenever F is not a linear forest, we clarify the complexity on F -free graphs for linear forests F with at most six vertices, except the P6. In particular, we show that the problem is efficiently solvable if F has at most five vertices. Efficient Edge Domination We provide a linear time algorithm for chordal bipartite graphs and an O(nm)-time algorithm for hole-free graphs, which solves the open question, whether Efficient Edge Domination is effi- ciently solvable on weakly chordal graphs. Monopolarity We formulate a graph property that enables the reduction from Monopolarity to 2-SAT and refine this idea to develop an efficient algorithm for a graph class that contains nearly all known efficiently solvable cases. Polarity Motivated by our result that Polarity is NP-complete on planar graphs, we introduce an algorithmic framework for subclasses of planar graphs. We show that the framework can be implemented efficiently for hole-free planar and maximal planar graphs. Acknowledgements First, I would like to thank my advisor Andreas Brandstädt for his support and for giving me the opportunity to work on this research in a friendly and warm atmosphere. Many thanks also to my colleagues Văn Bằng Lê and Christian Rosenke for having a sympathetic ear for all my scientific and non-scientific questions. I want to thank my coauthors Andreas, Bang, Christian, and Martin. It was a pleasure to work with you and I learned a lot from it. Finally, I want to express my gratitude to the patient proofreaders Robert Waltemath and Nicole Nevries. A very special thank to Christian Rosenke for finding the most hidden mistakes and giving me very helpful hints. Contents 1 Introduction 1 1.1 Preliminaries . .5 1.2 Efficient Domination . 12 1.3 Efficient Edge Domination . 16 1.4 Polarity . 19 2 Efficient Domination 25 2.1 Graphs Without Induced Linear Forests . 30 2.2 Hangings . 34 2.3 Maximum Weighted Independent Set . 56 3 Efficient Edge Domination 61 3.1 Chordal bipartite graphs . 65 3.2 Hole-free graphs . 70 4 Polarity and Monopolarity 85 4.1 Monopolar Extension via 2-Satisfiability . 87 4.2 A Superclass of Chair-free Graphs . 93 4.3 A Superclass of Hole-free Graphs . 98 4.4 Polarity on Planar Graphs . 109 5 NP-completeness Results 113 5.1 Reduction Framework . 113 5.2 Efficient Domination . 123 5.3 Efficient Edge Domination . 125 5.4 Polarity and Monopolarity . 127 6 Conclusion 133 Bibliography 134 1 Introduction One fundamental topic in theoretical computer science is the complexity analysis of computational problems, in particular the classification according to the classes P and NP. For practical problems, this classification is indicative for the existence or nonexistence of algorithms that solve the problems exactly and fast enough. Identifying problems that are in P or NP-complete is of high theoretical interest as well, since it is still unknown whether P and NP are really different. The presented research addresses this topic in the context of algorith- mic graph theory. We consider four graph problems, namely efficient domination, efficient edge domination, and polarity, as well as monopolarity, a special case of polarity. All these problems ask for a collection of objects that simultaneously is a packing and a covering. Objects of a packing have to be free of “conflicts”, while objects of a covering have to “span” all elements of another set. In optimization problems, we seek for largest possible packings and smallest possible coverings. A prominent example that asks for a collection that is a packing and a covering at the same time is the exact cover problem. It is one of Karp’s 21 eminent NP-complete problems published in 1972 [58]. For a given family of subsets F over a ground set S, it asks whether there is a subset F ? ⊆ F such that every element of S is contained in exactly one member of F ?. Notice that the packing part seeks for a subset of F such that its elements are pairwise disjoint and the covering part seeks for a subset of F such that the union of all its elements is S. Efficient domination, introduced in [6], is a special case of exact cover: Given a graph G, it asks if the family F of the closed neighborhoods of all vertices has a subset F ? that exactly covers the vertex set V . In literature, efficient domination is also expressed as finding an efficient dominating set, that is, an independent vertex subset D such that all other vertices have exactly one neighbor in D. Efficient edge domination, introduced in [49], is the edge-version of 2 CHAPTER 1. INTRODUCTION efficient domination. We seek an edge subset D such that no two of its edges share an endpoint and all other edges share exactly one endpoint with exactly one edge of D. Polarity, introduced in [91, 92], asks if the vertex set of a graph can be partitioned into a set that induces a complete multipartite graph and a set that induces a disjoint union of complete graphs. The packing aspect of polarity is the demand for a family of independent sets that are pairwise completely connected; the covering aspect requires that every induced path of length 3 has to share a vertex with at least one of these independent sets. Monopolarity is a well-studied special case of Polarity where the complete multipartite graph has to be edgeless. Hence, we seek for a partition into an edgeless graph and a disjoint union of complete graphs. Notice that efficient edge domination can be formulated as the question whether a graph can be partitioned into an edgeless graph and a 1-regular graph. Since 1-regular graphs are exactly the disjoint unions of complete graphs on two vertices, efficient edge domination is a special case of monopolarity. Efficient domination, efficient edge domination, polarity, and monopolarity are known to be NP-complete in general and even on very restricted graph classes, that is, when the input graph is supposed to fulfill a certain property. Hence, in the last years, great efforts were made to identify graph classes which allow efficient algorithms. The last three sections of the introduction try to give a comprehensive overview of the achieved results. We contribute to this research in both directions. On the one hand, we utilize the similarity of the mentioned problems to develop a reduction scheme for NP-completeness proofs. This scheme is directly applicable for efficient domination, efficient edge domination, and monopolarity and we show that the results for monopolarity can be transfered to polarity. We prove that • efficient domination and efficient edge domination are NP- complete on planar bipartite graphs with maximum degree at most 3 and girth at least g, for every fixed g, • polarity and monopolarity are NP-complete on planar triangle-free graphs with maximum degree at most 3, and • polarity and monopolarity are NP-complete on planar graphs with maximum degree at most 3 that contain no induced cycles of a length between 4 and g, for every fixed g ≥ 4. Since polarity is closed under taking complements, it also remains NP- 3 complete on the complements of the respective classes, including for example hole-free graphs. Notice that all problems are NP-complete on planar graphs and that the second result for polarity/monopolarity fulfills a weak form of a girth restriction. On the other hand, we present tractability results for all problems: Efficient domination Using known results, we point out that efficient domination is NP- complete on F -free graphs whenever F is not a linear forest. Consequently, we analyze the complexity on F -free graphs for linear forests F . We provide • a robust O(nm)-time algorithm for P5-free graphs, 2.38 • an O(min(nm, n ))-time algorithm for P5-free graphs, and • a robust O(nm)-time algorithm for (P4 + P2)-free graphs.